Convergence of Expensive Multi-Objective Optimizers: From ParEGO to ExTrEMO

Real-world multi-objective optimization problems often rely on physics-based simulators or physical experiments to assess solution quality, resulting in significant computational costs. In such scenarios, Gaussian process (GP) surrogate-assisted optimizers have demonstrated exceptional optimization performance. This article focuses on the theoretical convergence analysis of two existing decomposition-based GP-assisted optimizers: ParEGO with the upper confidence bound (ParEGO-UCB) and ExTrEMO. Unlike prior studies that typically assume a single weight vector for scalarization, this work primarily investigates multi-weight vector settings. Specifically, we analyze the regret bound of ParEGO-UCB within a rigorous theoretical framework and prove that, under a multi-weight vector setting, its convergence rate surpasses that of GP-UCB, which independently and sequentially optimizes multiple decomposed subproblems. Building on this foundation, we further explore the convergence properties of ExTrEMO, an expensive multi-objective optimizer designed for multi-source transfer optimization, in the context of multi-weight vector settings. Theoretical findings reveal that ExTrEMO achieves a tighter regret bound in multi-source settings compared to single-source scenarios, highlighting the advantages of leveraging additional sources to enhance optimization efficiency and convergence.